inductors and capacitors

Dave ShattuckUniversity of Houston

© University of Houston ECE 2300 Circuit Analysis

Dr. Dave ShattuckAssociate Professor, ECE Dept.

Lecture Set #11Inductors and Capacitors

[email protected] 743-4422

W326-D3

Lecture Set #11Inductors and Capacitors


© University of Houston

Overview of this Part Inductors and Capacitors

In this part, we will cover the following topics:

• Defining equations for inductors and capacitors

• Power and energy storage in inductors and capacitors

• Parallel and series combinations• Basic Rules for inductors and capacitor

s



Textbook Coverage

Approximately this same material is covered in your textbook in the following sections:

• Electric Circuits 7th Ed. by Nilsson and Riedel: Sections 6.1 through 6.3



Basic Elements, ReviewWe are now going to pick

up the remaining basic circuit elements that we will be covering in this course.



Circuit Elements

• In circuits, we think about basic circuit elements that are the basic “building blocks” of our circuits. This is similar to what we do in Chemistry with chemical elements like oxygen or nitrogen.

• A circuit element cannot be broken down or subdivided into other circuit elements.

• A circuit element can be defined in terms of the behavior of the voltage and current at its terminals.


© University of HoustonThe 5 Basic Circuit Elements

There are 5 basic circuit elements:1. Voltage sources2. Current sources3. Resistors4. Inductors5. CapacitorsWe defined the first three elements previously.

We will now introduce inductors or capacitors.



Inductors• An inductor is a two terminal circuit element that has a voltage across its terminals which is proportional to the derivative of the current through its terminals.

• The coefficient of this proportionality is the defining characteristic of an inductor.

• An inductor is the device that we use to model the effect of magnetic fields on circuit variables. The energy stored in magnetic fields has effects on voltage and current. We use the inductor component to model these effects.

In many cases a coil of wire can be modeled as an inductor.



• An inductor obeys the expression

where vL is the voltage across the inductor, and iL is the current through the inductor, and LX is called the inductance.

• In addition, it works both ways. If something obeys this expression, we can think of it, and model it, as an inductor.

• The unit ([Henry] or [H]) is named for Joseph Henry, and is equal to a [Volt-second/Ampere].

Inductors – Definition and Units

There is an inductance whenever we have magnetic fields produced, and there are magnetic fields whenever current flows. However, this inductance is often negligible except when we wind wires in coils to concentrate the effects.

LL X

div L

dt



LX= #[H]

vL -+

iL

Schematic Symbol for Inductors

The schematic symbol that we use for inductors is shown here.

This is intended to indicate that the schematic symbol can be labeled either with a variable, like LX, or a value, with some number, and units. An example might be 390[mH]. It could also be labeled with both.

LL X

div L

dt


© University of HoustonInductor Polarities

• Previously, we have emphasized the important of reference polarities of current sources and voltages sources. There is no corresponding polarity to an inductor. You can flip it end-for-end, and it will behave the same way.

• However, similar to a resistor, direction matters in one sense; we need to have defined the voltage and current in the passive sign convention to use the defining equation the way we have it here.

LL X

div L

dt



LX= #[H]

vL -+

iL

Passive and Active Sign Convention for Inductors

The sign of the equation that we use for inductors depends on whether we have used the passive sign convention or the active sign convention.

LL X

div L

dt

Passive Sign Convention

LX= #[H]

vL -+

iL

LL X

div L

dt

Active Sign Convention


© University of HoustonDefining Equation, Integral

Form, DerivationThe defining equation for the inductor,

can be rewritten in another way. If we want to express the current in terms of the voltage, we can integrate both sides. We get

LL X

div L

dt

0 0

( ) .t t

LL Xt t

div t dt L dt

dt

We pick t0 and t for limits of the integral, where t is time, and t0 is an arbitrary time value, often zero. The inductance, LX, is constant, and can be taken out of the integral. To avoid confusion, we introduce the dummy variable s in the integral. We get

0 0

1( ) .

t t

L Lt tX

v s ds diL

We finish the derivation in the next slide.



We can take this equation and perform the integral on the right hand side. When we do this we get

Thus, we can solve for iL(t), and we have two defining equations for the inductor,

Defining Equations for Inductors

.LL X

div L

dt

00

1( ) ( ) ( ),

t

L L LtX

i t v s ds i tL

and

0 0

1( ) .

t t

L Lt tX

v s ds diL

00

1( ) ( ) ( ).

t

L L LtX

v s ds i t i tL

Remember that both of these are defined for the passive sign convention for iL and vL. If not, then we need a negative sign in these equations.



Note 1The implications of these equations are significant. For

example, if the current is not changing, then the voltage will be zero. This current could be a constant value, and large, and an inductor will have no voltage across it. This is counter-intuitive for many students. That is because they are thinking of actual coils, which have some finite resistance in their wires. For us, an ideal inductor has no resistance; it simply obeys the laws below.

We might model a coil with both inductors and resistors, but for now, all we need to note is what happens with these ideal elements.

.LL X

div L

dt

00

1( ) ( ) ( ),

t

L L LtX

i t v s ds i tL

and



Note 2The implications of these equations are significant.

Another implication is that we cannot change the current through an inductor instantaneously. If we were to make such a change, the derivative of current with respect to time would be infinity, and the voltage would have to be infinite. Since it is not possible to have an infinite voltage, it must be impossible to change the current through an inductor instantaneously.

.LL X

div L

dt

00

1( ) ( ) ( ),

t

L L LtX

i t v s ds i tL

and



Note 3Some students are troubled by the introduction of the

dummy variable s in the integral form of this equation, below. It is not really necessary to introduce a dummy variable. It really doesn’t matter what variable is integrated over, because when the limits are inserted, that variable goes away.

.LL X

div L

dt

00

1( ) ( ) ( ),

t

L L LtX

i t v s ds i tL

and

The independent variable t is in the limits of the integral. This is indicated by the iL(t) on the left-hand side of the equation.

Remember, the integral here is not a function of s. It is a function of t.

This is a constant.


© University of Houston Energy in Inductors, DerivationWe can take the defining equation for the inductor, and use it to

solve for the energy stored in the magnetic field associated with the inductor. First, we note that the power is voltage times current, as it has always been. So, we can write,

.LL L L X L

dw dip v i L i

dt dt

Now, we can multiply each side by dt, and integrate both sides to get

0 0.

L Lw i

X L Ldw L i di Note, that when we integrated, we needed limits. We know that when the current is zero, there is no magnetic field, and therefore there can be no energy in the magnetic field. That allowed us to use 0 for the lower limits. The upper limits came since we will have the energy stored, wL, for a given value of current, iL. The derivation continues on the next slide.


© University of Houston Energy in Inductors, FormulaWe had the integral for the energy,

0 0.

L Lw i

X L Ldw L i di Now, we perform the integration. Note that LX is a constant, independent of the current through the inductor, so we can take it out of the integral. We have

2

0 0 .2L

L X

iw L

We simplify this, and get the formula for energy stored in the inductor,

21 .2L X Lw L i


© University of Houston Notes1.We took some mathematical liberties in this derivation. For example,

we do not really multiply both sides by dt, but the results that we obtain are correct here.

2.Note that the energy is a function of the current squared, which will be positive. We will assume that our inductance is also positive, and clearly ½ is positive. So, the energy stored in the magnetic field of an inductor will be positive.

3.These three equations are useful, and should be learned or written down.

Go back to Overview

slide.

212L X Lw L i

LL X

div L

dt0

0

1( ) ( ) ( )

t

L L LtX

i t v s ds i tL



Capacitors• A capacitor is a two terminal circuit element that has a current through its terminals which is proportional to the derivative of the voltage across its terminals.

• The coefficient of this proportionality is the defining characteristic of a capacitor.

• A capacitor is the device that we use to model the effect of electric fields on circuit variables. The energy stored in electric fields has effects on voltage and current. We use the capacitor component to model these effects.

In many cases the idea of two parallel conductive plates is used when we think of a capacitor, since this arrangement facilitates the production of an electric field.


© University of Houston Capacitors – Definition and Units

• An capacitor obeys the expression

where vC is the voltage across the capacitor, and iC is the current through the capacitor, and CX is called the capacitance.

• In addition, it works both ways. If something obeys this expression, we can think of it, and model it, as an capacitor.

• The unit ([Farad] or [F]) is named for Michael Faraday, and is equal to a [Ampere-second/Volt]. Since an [Ampere] is a [Coulomb/second], we can also say that a [F]=[C/V].

There is a capacitance whenever we have electric fields produced, and there are electric fields whenever there is a voltage between conductors. However, this capacitance is often negligible.

CC X

dvi C

dt



CX = #[F]

iC

vC+ -

Schematic Symbol for Capacitors

The schematic symbol that we use for capacitors is shown here.

This is intended to indicate that the schematic symbol can be labeled either with a variable, like CX, or a value, with some number, and units. An example might be 100[mF]. It could also be labeled with both.

CC X

dvi C

dt


© University of HoustonCapacitor Polarities

• Previously, we have emphasized the important of reference polarities of current sources and voltages sources. There is no corresponding polarity to an capacitor. For most capacitors, you can flip them end-for-end, and they will behave the same way. An exception to this rule is an electrolytic capacitor, which must be placed so that the voltage across it will be in the proper polarity. This polarity is usually marked on the capacitor.

• In any case, similar to a resistor, direction matters in one sense; we need to have defined the voltage and current in the passive sign convention to use the defining equation the way we have it here.

CC X

dvi C

dt


© University of Houston Passive and Active Sign Convention for Capacitors

The sign of the equation that we use for capacitors depends on whether we have used the passive sign convention or the active sign convention.

Passive Sign Convention Active Sign Convention

CC X

dvi C

dt C

C X

dvi C

dt

CX = #[F]

iC

vC+ -

CX = #[F]

iC

vC+ -



The defining equation for the capacitor,

can be rewritten in another way. If we want to express the voltage in terms of the current, we can integrate both sides. We get

Defining Equation, Integral Form, Derivation

0 0

( ) .t t

CC Xt t

dvi t dt C dt

dt

We pick t0 and t for limits of the integral, where t is time, and t0 is an arbitrary time value, often zero. The capacitance, CX, is constant, and can be taken out of the integral. To avoid confusion, we introduce the dummy variable s in the integral. We get

0 0

1( ) .

t t

C Ct tX

i s ds dvC

We finish the derivation in the next slide.

CC X

dvi C

dt


© University of HoustonDefining Equations for

Capacitors

We can take this equation and perform the integral on the right hand side. When we do this we get

Thus, we can solve for vC(t), and we have two defining equations for the capacitor,

.CC X

dvi C

dt

00

1( ) ( ) ( ),

t

C C CtX

v t i s ds v tC

and

0 0

1( ) .

t t

C Ct tX

i s ds dvC

00

1( ) ( ) ( ).

t

C C CtX

i s ds v t v tC

Remember that both of these are defined for the passive sign convention for iC and vC. If not, then we need a negative sign in these equations.



Note 1The implications of these equations are significant. For

example, if the voltage is not changing, then the current will be zero. This voltage could be a constant value, and large, and a capacitor will have no current through it.

For many students this is easier to accept than the analogous case with the inductor. This is because practical capacitors have a large enough resistance of the dielectric material between the capacitor plates, so that the current flow through it is generally negligible.

.CC X

dvi C

dt

00

1( ) ( ) ( ),

t

C C CtX

v t i s ds v tC

and



Note 2

The implications of these equations are significant. Another implication is that we cannot change the voltage across a capacitor instantaneously. If we were to make such a change, the derivative of voltage with respect to time would be infinity, and the current would have to be infinite. Since it is not possible to have an infinite current, it must be impossible to change the voltage across a capacitor instantaneously.

.CC X

dvi C

dt

00

1( ) ( ) ( ),

t

C C CtX

v t i s ds v tC

and



.CC X

dvi C

dt

00

1( ) ( ) ( ),

t

C C CtX

v t i s ds v tC

and

Note 3Some students are troubled by the introduction of the

dummy variable s in the integral form of this equation, below. It is not really necessary to introduce a dummy variable. It really doesn’t matter what variable is integrated over, because when the limits are inserted, that variable goes away.

The independent variable t is in the limits of the integral. This is indicated by the vC(t) on the left-hand side of the equation.

Remember, the integral here is not a function of s. It is a function of t.

This is a constant.


© University of HoustonEnergy in Capacitors,

DerivationWe can take the defining equation for the capacitor, and use it

to solve for the energy stored in the electric field associated with the capacitor. First, we note that the power is voltage times current, as it has always been. So, we can write,

.CC C C C X

dvdwp v i v C

dt dt

Now, we can multiply each side by dt, and integrate both sides to get

0 0.

C Cw v

X C Cdw C v dv Note, that when we integrated, we needed limits. We know that when the voltage is zero, there is no electric field, and therefore there can be no energy in the electric field. That allowed us to use 0 for the lower limits. The upper limits came since we will have the energy stored, wC, for a given value of voltage, vC. The derivation continues on the next slide.



Energy in Capacitors, FormulaWe had the integral for the energy,

0 0.

C Cw v

X C Cdw C v dv Now, we perform the integration. Note that CX is a constant, independent of the voltage across the capacitor, so we can take it out of the integral. We have 2

0 0 .2C

C X

vw C

We simplify this, and get the formula for energy stored in the capacitor,

21 .2C X Cw C v


© University of Houston Notes1.We took some mathematical liberties in this derivation. For example,

we do not really multiply both sides by dt, but the results that we obtain are correct here.

2.Note that the energy is a function of the voltage squared, which will be positive. We will assume that our capacitance is also positive, and clearly ½ is positive. So, the energy stored in the electric field of an capacitor will be positive.

3.These three equations are useful, and should be learned or written down.

Go back to Overview

slide.

21 .2C X Cw C v

.CC X

dvi C

dt

00

1( ) ( ) ( ),

t

C C CtX

v t i s ds v tC



Series Inductors Equivalent CircuitsTwo series

inductors, L1 and L2, can be replaced with an equivalent circuit with a single inductor LEQ, as long as

1 2.EQL L L

L2

L1

Rest ofthe

Circuit

Rest ofthe

CircuitLEQ



More than 2 Series Inductors

This rule can be extended to more than two series inductors. In this case, for N series inductors, we have

1 2 ... .EQ NL L L L L2

L1

Rest ofthe

Circuit

Rest ofthe

CircuitLEQ


© University of HoustonSeries Inductors Equivalent

Circuits: A ReminderTwo series inductors, L1 and L2, can be replaced with an equivalent circuit with a single inductor LEQ, as long as

1 2.EQL L L

Remember that these two equivalent circuits are equivalent only with respect to the circuit connected to them. (In yellow here.)

L2

L1

Rest ofthe

Circuit

Rest ofthe

CircuitLEQ


© University of HoustonSeries Inductors Equivalent

Circuits: Initial ConditionsTwo series inductors, L1 and L2, can be replaced with an equivalent circuit with a single inductor LEQ, as long as

1 2.EQL L L

To be equivalent with respect to the “rest of the circuit”, we must have any initial condition be the same as well. That is, iL1(t0) must equal iLEQ(t0).

L2

L1

Rest ofthe

Circuit

Rest ofthe

CircuitLEQ

iL1(t) iLEQ(t)



Parallel Inductors Equivalent CircuitsTwo parallel

inductors, L1 and L2, can be replaced with an equivalent circuit with a single inductor LEQ, as long as

1 2

1 2

1 2

1 1 1, or

.

EQ

EQ

L L L

L LL

L L

L2L1

Rest ofthe

Circuit

Rest ofthe

CircuitLEQ

iL1(t)

iLEQ(t)

iL2(t)


© University of Houston More than 2 Parallel Inductors

This rule can be extended to more than two parallel inductors. In this case, for N parallel inductors, we have

1 2

1 1 1 1... .

EQ NL L L L

L2L1

Rest ofthe

Circuit

Rest ofthe

CircuitLEQ

iL1(t)

iLEQ(t)

iL2(t)

The product over sum rule only works for two inductors.


© University of Houston Parallel Inductors Equivalent Circuits: A Reminder


L2L1

Rest ofthe

Circuit

Rest ofthe

CircuitLEQ

iL1(t)

iLEQ(t)

iL2(t)

Two parallel inductors, L1 and L2, can be replaced with an equivalent circuit with a single inductor LEQ, as long as

1 2

1 2

1 2

1 1 1, or

.

EQ

EQ

L L L

L LL

L L


© University of HoustonParallel Inductors Equivalent

Circuits: Initial Conditions

• To be equivalent with respect to the “rest of the circuit”, we must have any initial condition be the same as well. That is,

L2L1

Rest ofthe

Circuit

Rest ofthe

CircuitLEQ

iL1(t)

iLEQ(t)

iL2(t)

0 1 0 2 0( ) ( ) ( ).LEQ L Li t i t i t


© University of HoustonParallel Capacitors Equivalent Circuits

Two parallel capacitors, C1 and C2, can be replaced with an equivalent circuit with a single capacitor CEQ, as long as

1 2.EQC C C Rest of

theCircuit

Rest ofthe

Circuit

C1 C2CEQ



Rest ofthe

Circuit

Rest ofthe

Circuit

C1 C2CEQ

More than 2 Parallel Capacitors

This rule can be extended to more than two parallel capacitors. In this case, for N parallel capacitors, we have

1 2 ... .EQ NC C C C



Rest ofthe

Circuit

Rest ofthe

Circuit

C1 C2CEQ

Parallel Capacitors Equivalent Circuits: A Reminder

This rule can be extended to more than two parallel capacitors. In this case, for N parallel capacitors, we have


1 2 ... .EQ NC C C C


© University of Houston Parallel Capacitors Equivalent Circuits: Initial ConditionsTwo parallel

capacitors, C1 and C2, can be replaced with an equivalent circuit with a single inductor CEQ, as long as

1 2.EQC C C

To be equivalent with respect to the “rest of the circuit”, we must have any initial condition be the same as well. That is, vC1(t0) must equal vCEQ(t0).

Rest ofthe

Circuit

Rest ofthe

Circuit

C1 C2CEQ

vC1(t)

-

+vCEQ(t)

-

+


© University of Houston Series Capacitors Equivalent CircuitsTwo series

capacitors, C1 and C2, can be replaced with an equivalent circuit with a single inductor CEQ, as long as

1 2

1 2

1 2

1 1 1, or

.

EQ

EQ

C C C

C CC

C C

C2

C1

Rest ofthe

Circuit

Rest ofthe

CircuitCEQ


© University of Houston More than 2 Series Capacitors

This rule can be extended to more than two series capacitors. In this case, for N series capacitors, we have

1 2

1 1 1 1... .

EQ NC C C C

The product over sum rule only works for two capacitors.

C2

C1

Rest ofthe

Circuit

Rest ofthe

CircuitCEQ



Series Capacitors Equivalent Circuits: A Reminder

Two series capacitors, C1 and C2, can be replaced with an equivalent circuit with a single capacitor CEQ, as long as

1 2

1 2

1 2

1 1 1, or

.

EQ

EQ

C C C

C CC

C C

C2

C1

Rest ofthe

Circuit

Rest ofthe

CircuitCEQ



© University of HoustonSeries Capacitors Equivalent

Circuits: Initial Conditions

• To be equivalent with respect to the “rest of the circuit”, we must have any initial condition be the same as well. That is,

0 1 0 2 0( ) ( ) ( ).CEQ C Cv t v t v t

C2

C1

Rest ofthe

Circuit

Rest ofthe

CircuitCEQ

vC1(t)

-

+

vC2(t)

-

+

vCEQ(t)

-

+



Inductor Rules and Equations• For inductors,

we have the following rules and equations which hold:

0

0

2

( )1: ( )

12 : ( ) ( ) ( )

13: ( ) ( )24: No instantaneous change in current through the inductor.

5: When there is no change in the current, there is no voltage.

6: Appears as

LL X

t

L L LtX

L X L

di tv t L

dt

i t v s ds i tL

w t L i t

a short-circuit at dc.

LX= #[H]

vL -+

iL


© University of HoustonInductor Rules and Equations

– dc Note• For

inductors, we have the following rules and equations which hold:

0

0

2

( )1: ( )

12 : ( ) ( ) ( )

13: ( ) ( )24: No instantaneous change in current through the inductor.

5: When there is no change in the current, there is no voltage.

6: Appears as

LL X

t

L L LtX

L X L

di tv t L

dt

i t v s ds i tL

w t L i t

a short-circuit at dc.

LX= #[H]

vL -+

iL The phrase dc may be new to some students. By “dc”, we mean that nothing is changing. It came from the phrase “direct current”, but is now used in many additional situations, where things are constant. It is used with more than just current.


© University of HoustonCapacitor Rules and

Equations• For capacitors,

we have the following rules and equations which hold:

0

0

2

( )1: ( )

12 : ( ) ( ) ( )

13: ( ) ( )24: No instantaneous change in voltage across the capacitor.

5: When there is no change in the voltage, there is no current.

6: Appears as

CC X

t

C C CtX

C X C

dv ti t C

dt

v t i s ds v tC

w t C v t

a open-circuit at dc.

CX = #[F]

iC

vC+ -


© University of HoustonWhy do we cover inductors?

Aren’t capacitors good enough for everything?

• This is a good question. Capacitors, for practical reasons, are closer to ideal in their behavior than inductors. In addition, it is easier to place capacitors in integrated circuits, than it is to use inductors. Therefore, we see capacitors being used far more often than we see inductors being used.

• Still, there are some applications where inductors simply must be used. Transformers are a case in point. When we find these applications, we should be ready, so that we can handle inductors.

Go back to Overview

slide.

inductors and capacitors

Education